Common Factor Simplification

Question

This is the problem I currently have.

$$4(3x+2)^3 3(x+5)^{-3}-3(x+5)^{-4}(3x+2)^4$$

The text book gets the following:

$$3(3x+2)^3(x+5)^{-4}[4(x+5)-(3x+2)]$$

Which simplifies to the below:

$$3(3x+2)^3(x+5)^{-4}(x+18)$$

But I don't understand how they got to that solution.

Specifically how they got to second step $$[4(x+5)-(3x+2)]$$

Answer 1

They used identity $b^{-3} = b^{-4} \cdot b$ (for $b \neq 0$).

Specifically, let $a = 3x+2$ and $b = x + 5$

Then $$4a^3\cdot3b^{-3}-3a^4b^{-4} = 3a^3b^{-4}(4b-a)$$. You can convince yourself by opening parenthesis, that this is indeed equality.

Answer 2

\begin{align*} &\;\,\text{By the laws of exponents,}\\[8pt] &(3x+2)^{4}=(3x+2)^{3}(3x+3)\\[2pt] &(x+5)^{-3}=(x+5)^{-4}(x+5)\\[8pt] &\;\text{hence}\\[8pt] &4(3x+2)^33(x+5)^{-3}-3(x+5)^{-4}(3x+2)^4\tag{1}\\[4pt] =\;\,&4(3x+2)^33\bigl((x+5)^{-4}(x+5)\bigr)-3(x+5)^{-4}\bigl((3x+2)^3(3x+2)\bigr)\tag{2}\\[4pt] =\;\,&\bigl(3(3x+2)^3(x+5)^{-4}\bigr)\bigl(4(x+5)\bigr)-\bigl(3(3x+2)^3(x+5)^{-4}\bigr)(3x+2)\tag{3}\\[4pt] =\;\,&\bigl(3(3x+2)^3(x+5)^{-4}\bigr)\bigl(4(x+5)-(3x+2)\bigr)\tag{4}\\[4pt] =\;\,&\,\text{. . .}\\[4pt] \end{align*}

The basic strategy for these kinds of problem is to factor out the least power of the common terms. In this case, the common terms are $(3x+2)$ and $(x+5)$, so you want to factor out $(3x+2)^3$ (since the exponent $3$ is less than the exponent $4$), and you also want to factor out $(x+5)^{-4}$ (since the exponent $-4$ is less than the exponent $-3$).

Note:

This is the same as dEmigOd's answer, but without explicitly replacing $3x+2$ by $a$, and $x+5$ by $b$. If you make explicit substitutions, the factorization is more obvious, but if instead you do the substitutions "mentally" (i.e., visualize $(3x+2)$ and $(x+5)$ as "blocks"), the factorization can be explained using the steps shown above.

Which is better? It's a matter of taste. Personally, if I can't visualize the blocks easily, I make explicit substitutions (making sure to reverse the substitutions at the end). For this problem, the visualization is not that hard, so I'd probably do it essentially as shown above, but I'd only show steps $(1)$ and $(4)$, regarding the omitted steps $(2)$ and $(3)$ as "implied".

Answer 3

To reach the 2nd step just take common from both terms 3(3x+2)^(3 ) (x+5)^(-4) If it seems difficult then first multiply and divide the whole expression with the term (x+5)